Global phase portraits of kukles differential systems with homogeneous polynomial nonlinearities of degree 6 having a center and their small limit cycles

© 2016 World Scientific Publishing Company. We provide the nine topological global phase portraits in the Poincaré disk of the family of the centers of Kukles polynomial differential systems of the form x. = -y, = x + ax5y + bx3y3 + cxy5, where x,y and a,b,c are real parameters satisfying a2 + b2 + c2=0. Using averaging theory up to sixth order we determine the number of limit cycles which bifurcate from the origin when we perturb this system first inside the class of all homogeneous polynomial differential systems of degree 6, and second inside the class of all polynomial differential systems of degree 6.